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Home , 20 Massalia, 24 Themis, 419 Aurelia, 44 Nysa, 55 Pandora, 64 Angelina

Meteoritics & 44, Nr 12, 1937–1946 (2009) Abstract available online at http://meteoritics.org

Asteroid photometric and polarimetric phase curves: Joint linear-exponential modeling

K. MUINONEN1, 2, A. PENTTILÄ1, A. CELLINO3, I. N. BELSKAYA4, M. DELBÒ5, A. C. LEVASSEUR-REGOURD6, and E. F. TEDESCO7

1University of Helsinki, Observatory, Kopernikuksentie 1, P.O. BOX 14, FI-00014 U. Helsinki, Finland 2Finnish Geodetic Institute, Geodeetinrinne 2, P.O. Box 15, FI-02431 Masala, Finland 3INAF-Osservatorio Astronomico di Torino, strada Osservatorio 20, 10025 Pino Torinese, Italy 4Astronomical Institute of Kharkiv National University, 35 Sumska Street, 61035 Kharkiv, Ukraine 5IUMR 6202 Laboratoire Cassiopée, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice, Cedex 4, France 6UPMC Univ. Paris 06, UMR 7620, BP3, 91371 Verrières, France 7Planetary Science Institute, 1700 E. Ft. Lowell Road, Tucson, Arizona 85719, USA *Corresponding author. E-mail: [email protected] (Received 01 April, 2009; revision accepted 18 August 2009)

Abstract–We present Markov-Chain Monte-Carlo methods (MCMC) for the derivation of empirical model parameters for photometric and polarimetric phase curves of . Here we model the two phase curves jointly at phase angles գ25° using a linear-exponential model, accounting for the effect in disk-integrated brightness and the negative branch in the degree of linear polarization. We apply the MCMC methods to V-band phase curves of asteroids 419 Aurelia (taxonomic class F), 24 (C), 1 (G), (S), 55 (M), and (E). We show that the photometric and polarimetric phase curves can be described using a common nonlinear parameter for the angular widths of the opposition effect and negative-polarization branch, thus supporting the hypothesis of common physical mechanisms being responsible for the phenomena. Furthermore, incorporating polarimetric observations removes the indeterminacy of the opposition effect for 1 Ceres. We unveil a trend in the interrelation between the enhancement factor of the opposition effect and the angular width: the enhancement factor decreases with decreasing angular width. The minimum polarization and the polarimetric slope at the inversion angle show systematic trends when plotted against the angular width and the normalized photometric slope parameter. Our new approach allows improved analyses of possible similarities and differences among asteroidal surfaces.

INTRODUCTION (Harris et al. 1989b; Rosenbush 2009), whereas in some cases negative polarization, in particular, extends over a wide range Two ubiquitous phenomena are observed for asteroids in (Cellino et al. 2006). and other atmosphereless solar system objects as well as for Coherent backscattering, single scattering, and cometary and interplanetary dust near opposition: a negative shadowing have been considered as mechanisms responsible branch in the degree of linear polarization (“negative for the phenomena (see review in Muinonen et al. 2002a). polarization”) and a nonlinear enhancement of brightness Coherent backscattering is an interference mechanism that (opposition effect). Negative polarization refers to the case can contribute to both brightness and polarization (Hapke where the scattered intensity component parallel to the - 1990; Muinonen 1989, 1990; Shkuratov 1985, 1988, 1989; object-observer plane (scattering plane) predominates over Mishchenko and Dlugach 1993), single-scattering the one perpendicular to the plane. The negative-polarization interference effects can contribute to both phenomena over a and opposition-effect phenomena are constrained to Sun- wider range of phase angles (e.g., Muinonen et al. 2007; object-observer angles (phase angles) of գ25° and գ10°, Tyynelä et al. 2007) whereas shadowing is thought to respectively. In some cases, the phenomena show up at contribute to the opposition effect only (see Muinonen et al. extremely small phase angles within a degree from opposition 2002a).

1937 © The Meteoritical Society, 2009. Printed in USA. 1938 K. Muinonen et al.

We analyze the photometric and polarimetric phase photometric phase curves must be constrained to positive curves jointly by using an empirical model developed to fit values at all phase angles. Nevertheless, the linear- the observations within the observational errors given. exponential model is suitable for the present study at small Empirical models are useful in planning further observations phase angles. of the objects and in grouping the objects based on the In order to distinguish between the photometric and similarities and differences in their phase curves. Note the polarimetric parameters, subscripts I (for intensity) and P (for important role played by the H, G system (Bowell polarization) are attached to the symbols a, b, k, and d. For et al. 1989) for research at large, and the ongoing , the empirical parameters are the amplitude aI and efforts to improve the system (Muinonen et al. 2008). angular width dI of the opposition effect, the background The present study emerges from the earlier ones for brightness bI, and the slope kI. By normalization at zero phase estimating the parameters of the brightness opposition effect angle, the number of parameters decreases to three. For (e.g., Bowell et al. 1989; Lumme et al. 1993; Belskaya and polarimetry, the empirical parameters are the amplitude aP Shevchenko 2000; Muinonen et al. 2002b; Rosenbush et al. and angular width dP of the negative-polarization branch, the 2002; Kaasalainen et al. 2003; Avramchuk et al. 2007) and of balancing amplitude bP, and the slope kP. Based on the physics = the negative polarization (e.g., Muinonen et al. 2002b; of light scattering, we assume bP aP so that the degree of Rosenbush et al. 2002; Kaasalainen et al. 2003; Lumme and linear polarization is zero at zero phase angle. For the joint = = Muinonen 2003; Levasseur-Regourd 2003, 2004). We seek linear-exponential modeling, we assume d dI dP so that d answers to the question whether it is possible to explain the is the single common nonlinear parameter in the photometric and polarimetric phase curves with a joint interpretation of the photometric and polarimetric phase empirical model involving common parameters for curves. photometry and polarimetry. Here we present the results of There are additional dependent parameters that are of studies at phase angles գ25 °. We provide practical Markov- interest in comparative studies. For the opposition effect, such Chain Monte-Carlo methods (MCMC) for obtaining reliable parameters are, e.g., the enhancement factor ζ and the angular error estimates for nonlinear model parameters. Note that half-width at half-maximum d1 , ---I Penttilä et al. (2005) made use of MCMC methods in their 2 a + b statistical analyses of asteroidal and cometary polarization ζ = ------I I , phase curves using the trigonometric (Lumme and Muinonen bI 2003) and polynomial models (Levasseur-Regourd 2004). = In the Theoretical and Numerical Methods section, we d1 dI ln 2. (2) ---I summarize the linear-exponential model for the photometric 2 and polarimetric applications and describe the MCMC For the negative polarization branch, the phase angle methods for sampling the parameters. Section 3 includes the αmin and value of minimum polarization Pmin are of special application of the methods to the V-band photometric and interest: polarimetric phase curves of asteroids 419 Aurelia k d (taxonomic class F), 24 Themis (C), 1 Ceres (G), 20 Massalia α ⎛⎞P P min = –dP ln ------, (S), (M), and 64 Angelina (E). We close the paper ⎝⎠aP with conclusions and future prospects. ⎛⎞kPdP THEORETICAL AND NUMERICAL METHODS Pmin = kPdP 1 – ln ------+ bP .(3) ⎝⎠aP Linear-Exponential Modeling The inversion angle of polarization α0 needs to be solved from an implicit equation (using, e.g., Newton’s method): We start with the four-parameter empirical linear- exponential model for the photometric and polarimetric phase α a exp⎛⎞–-----0- ++b k α = 0. (4) curves close to the opposition (Muinonen et al. 2002b; P ⎝⎠d P P 0 Kaasalainen et al. 2003): P ′ α The corresponding first derivative P0 gives the slope of f (α) = a exp ⎛⎞–--- + b + kα, (1) ⎝⎠d the polarization curve at the inversion angle which is known to correlate with the geometric and is used to derive where α is the phase angle, a, b, and k are the three linear reliable diameter estimates (see Cellino et al. 1999 and, in parameters, and d is the single nonlinear parameter. Note that particular, Belskaya et al. 2009 for wavelength dependences): the linear-exponential model does not describe polarimetric α and photometric phase curves at phase angles >30°: the ′ bP ++kP 0 kPdP P0 = ------.(5) polarimetric phase curves show maxima near 90° and the dP Asteroid photometric and polarimetric phase curves 1939

Markov-Chain Monte-Carlo Methods RESULTS AND DISCUSSION

The photometric and polarimetric phase curves are The photometric and polarimetric phase curves of analyzed using techniques developed further from those in asteroids 419 Aurelia (class F), 24 Themis (C), 1 Ceres (G), Muinonen et al. (2002b) and Kaasalainen et al. (2003). The a 20 Massalia (S), 55 Pandora (M), and 64 Angelina (E) in the posteriori probability density function (p.d.f.) for the V band are shown in Figs. 1 and 2, respectively, with the parameters P = (a, d, b, k) (for models separate for corresponding references and additional information on the photometry and polarimetry) is obtained from observations summarized in Table 1. We have chosen these asteroids for the present study because their phase curves are 1 2 p()P ∝ exp –---χ ()P ,(6)representative of the types of phase curves observed for 2 asteroids. Note, however, that there are still only small where χ2 denotes the square form composed of the matrix numbers of asteroids with sufficient phase-curve coverage in product of the O-C residuals (observed minus computed) and both photometry and polarimetry. the covariance matrix for the random observational errors. In For the individual photometric phase curves, all the error what follows, for simplicity, the covariance matrix is assumed standard deviations of the magnitude points are assumed to be to be diagonal. the same; and, for the individual polarimetric phase curves, all In earlier works (Muinonen et al. 2002b; Kaasalainen the error standard deviations of the polarization points are et al. 2003), we have additionally included Jeffreys’ a priori assumed to be the same. The error standard deviations are p.d.f. (Jeffreys 1948) as a multiplying p.d.f. on the right-hand determined as follows. For each individual photometric phase side of Equation 6. Jeffreys’ p.d.f is, in essence, a Jacobian curve, a best-fit four-parameter linear-exponential model is which guarantees that the probabilistic interpretation is derived using equal weights for the observations. The rms invariant in transformations from one nonlinear parameter set value of the fit is then chosen to be the error standard to another. We omit Jeffreys’ p.d.f. due to two main reasons: deviation for all the observations of that specific . first, it would introduce an additional indirect dependence on An analogous procedure is carried out for each polarimetric the observations; second, it would render the modeling phase curve. We have followed this choice because of either disproportionately complicated for the present study. missing detailed information on the errors (for photometry) or We sample the a posteriori p.d.f. p(P) in Equation 6 using somewhat inhomogeneous error estimates (for polarimetry) the classical Metropolis-Hastings MCMC method (Gilks and stress that the main conclusions of the study are not et al. 1996). For the so-called proposal p.d.f. q(P, Pj) centered affected by the choice of the error model. σ % at Pj with the proposed new value P, a new set of parameters In Figs. 1 and 2, we show the 3 or 99.7 error = is accepted as Pj + 1 P if, for a random deviate y ∈ [0,1], envelopes as obtained via MCMC with 50000 differing parameter sets (with varying numbers of repetitions for each p()P q()PP, < j y ------()(), .(7) set) in a joint treatment of photometry and polarimetry. At any p Pj q Pj P given phase angle, the envelope is constructed from the = Otherwise Pj + 1 Pj. If q is symmetric, the acceptance minimum and maximum values as given by the sample model criterion reduces to parameters belonging to the 3σ regime of the parameters. Outside the phase-angle ranges covered by the observations p()P y < ------. (8) but still within the phase-angle range of applicability for the p()P j linear-exponential model, the best-fit models and the In the present study, we utilize symmetric univariate envelopes serve as realistic predictors for the photometric and Gaussian proposal p.d.f.’s for each of the elements in polarimetric phase curves, although the purely empirical q(P, Pj). It can be shown that, under quite general conditions, character of the models limits their predictive value. the chain P1, P2, P3, . . . approaches to sample from the The MCMC method works efficiently in the joint correct distribution for P. treatment of photometry and polarimetry, with a delivery of For the joint empirical modeling, the six free parameters 50000 samples within a few CPU seconds on a modern are P = (d, aI, bI, kI, aP, k P) and, following the assumption of personal computer. This is a remarkable improvement in Gaussian observational errors, we define speed as compared to the earlier p.d.f. characterizations for the linear-exponential model separately for photometry and polarimetry (e.g., Muinonen et al. 2002b; Kaasalainen et al. χ2 χ2 χ2 = I + P,(9)2003). However, when comparing the computational speed to that of the earlier work, it is important to bear in mind that χ2 χ2 χ2 where I and P are the values of the photometric and Jeffreys’ a priori p.d.f. (Jeffreys 1943), excluded currently but polarimetric fits, respectively, with the two data sets taken to included earlier, commonly introduces a computational have uncorrelated observational errors. burden. Note that the so-called burn-in phase in MCMC 1940 K. Muinonen et al.

Fig. 1. Best-fit linear-exponential models (dashed lines) with MCMC-based 3σ (99.7%) error envelopes (solid lines) for the photometric observations of the following asteroids: a) 419 Aurelia (taxonomic class F), b) 24 Themis (C), c) 1 Ceres (G), d) 20 Massalia (S), e) 55 Pandora (M), and f) 64 Angelina (E). See Table 1 for additional information on and references to the observations. Asteroid photometric and polarimetric phase curves 1941

Fig. 2. As in Fig. 1 for the polarimetric observations. 1942 K. Muinonen et al.

Table 1. Asteroids utilized in the joint linear-exponential modeling of the photometric (label I ) and polarimetric phase curves (P). We show the number of observations Nobs, the minimum and maximum phase angles of the observations α< and α ° σ % > ( ), the error standard deviations fit for the photometric (in mag) and polarimetric observations (in ) and the references to the observations. σ Asteroid Nobs α< α> fit References I 419 Aurelia 13 0.11 14.2 0.014 Belskaya et al. 2002 24 Themis 22 0.34 20.8 0.010 Harris et al. 1989a 1 Ceres 16 1.73 19.2 0.018 Tedesco et al. 1983 20 Massalia 14 0.09 21.4 0.023 Gehrels 1956; Belskaya et al. 2003 55 Pandora 15 0.3 16.3 0.015 Shevchenko et al. 1993 64 Angelina 11 0.11 22.5 0.011 Harris et al. 1989a P 419 Aurelia 13 0.5 18.0 0.21 Belskaya et al. 2005 24 Themis 11 1.3 19.1 0.11 Chernova et al. 1994 1 Ceres 29 1.11 23.1 0.071 Zellner et al. 1974; Zellner and Gradie 1976 20 Massalia 13 0.08 23.2 0.089 Belskaya et al. 2003; Zellner and Gradie 1976 55 Pandora 9 0.3 15.2 0.14 Lupishko et al. 1994 64 Angelina 18 0.75 18.4 0.071 Zellner and Gradie 1976; Rosenbush et al. 2005

Table 2. Parameters of joint linear-exponential photometric and polarimetric models for the asteroids in Table 1 with MCMC-based two-sided error estimates that give the minimum and maximum deviations of each parameter within the top 3σ or 99.7% fraction of parameter sets sampled. We show the rms-values of the photometric and polarimetric fits, ζ = + angular width d, normalized opposition-effect amplitude aI/bI (the enhancement factor 1 aI/bI), normalized photometric slope kI/bI, polarimetric amplitude aP, and the polarimetric slope kP. 419 Aurelia 24 Themis 1 Ceres 20 Massalia 55 Pandora 64 Angelina rms (mag) 0.017 0.012 0.019 0.023 0.015 0.011 rms (%) 0.24 0.17 0.071 0.095 0.15 0.072 d (°) 10.5 6.71 6.77 2.59 1.87 1.00 −4.4 −2.0 −0.48 −0.69 −0.72 −0.28 +5.7 +2.2 +1.46 +1.22 +1.48 +0.48 aI /bI 0.94 0.68 0.51 0.55 0.352 0.317 −0.44 −0.22 −0.20 −0.13 −0.083 −0.055 +0.06 +0.32 +0.34 +0.22 +0.141 +0.054 ° −1 − − − − − − kI /bI (( ) ) 0.0157 0.0160 0.0191 0.0159 0.0222 0.0149 −0.0120 −0.0038 −0.0040 −0.0030 −0.0031 −0.0017 +0.0061 +0.0059 +0.0069 +0.0042 +0.0052 +0.0018 aP (%) 8.92 4.84 5.86 1.15 1.32 0.42 −4.3 −1.4 −0.26 −0.26 −0.40 −0.11 +8.3 +2.1 +1.08 +0.45 +0.56 +0.13 ° −1 kP (%( ) ) 0.49 0.222 0.3016 0.059 0.048 0.023 −0.17 −0.072 −0.0091 −0.018 −0.037 −0.012 +0.28 +0.095 +0.0418 +0.026 +0.052 +0.010 sampling, the computational phase where the Markov chain values of the fits using the joint model are in accordance with approaches the regime of higher p.d.f. values, is here absent the overall variation among the data points for each asteroid. since the chains are initiated with parameter values already Also shown in Table 2 are the best-fit model parameters resulting in good fits to the observations. together with their two-sided error estimates based on the Table 2 shows the rms-values of the photometric and maximum extent of each parameter within the 3σ regime. For polarimetric fits. Comparing the rms-values to the assumed an example of MCMC sampling, for asteroid 1 Ceres, the standard deviations of the observational errors tabulated in standard deviations of the Gaussian proposal p.d.f.’s for d, aI, ° ° −1 Table 1 shows a modest deterioration of the fits for a number bI, kI, aP, and kP are 0.03 , 0.003, 0.0001, 0.000007( ) , 0.03, of objects when the photometric and polarimetric and 0.0002(°)−1, respectively. The full Markov chain consists observations are treated jointly. This is the price we are of 222688 entries, resulting in the required 50000 different paying for reducing the number of parameters and using a samples and an efficiency of 22%. Note that a purely simple empirical model instead of a physical or more evolved photometric modeling of 1 Ceres would result in complete empirical model. A study of the statistical significance of the indeterminacy insofar as the angular width of the opposition deterioration is left for future. We only note that the rms effect is concerned. The indeterminacy is removed by the Asteroid photometric and polarimetric phase curves 1943

Fig. 3. Opposition-effect and negative-polarization parameters against the angular width d for the six asteroids studied (Table 1): a. enhancement factor ζ, b. normalized photometric slope k /b , c. minimum polarization P , and d. polarimetric slope P′ . Also shown are I I ′ min 0 parameters against the normalized photometric slope kI/bI: e. Pmin and f. P0. 1944 K. Muinonen et al.

Table 3. As in Table 2 for the phase angle of minimum polarization min, minimum polarization Pαmin, inversion angle of α ′ polarization 0, and the polarimetric slope at the inversion angle P 0. 419 Aurelia 24 Themis 1 Ceres 20 Massalia 55 Pandora 64 Angelina α ° min ( ) 5.70 7.91 7.19 5.24 5.0 2.88 −0.70 −0.77 −0.30 −0.83 −1.5 −0.62 +0.95 +0.80 +0.42 +1.21 +2.5 +0.93 % − − − − − − Pmin ( ) 0.92 1.59 1.66 0.69 0.99 0.328 −0.31 −0.23 −0.067 −0.15 −0.26 −0.097 +0.28 +0.26 +0.079 +0.14 +0.24 +0.077 α ° 0 ( ) 12.7 20.8 18.10 19.6 27.5 17.9 −1.4 −1.4 −0.38 −2.3 −9.8 −3.5 +1.8 +2.3 +0.43 +4.1 +6.5 +10.3 ′ % ° −1 P0 ( ( ) ) 0.239 0.189 0.2417 0.059 0.048 0.023 −0.057 −0.050 −0.0081 −0.018 −0.037 −0.012 +0.063 +0.048 +0.0178 +0.024 +0.052 +0.010 joint treatment of the photometric and polarimetric data sets. be located towards the left in the corresponding error domain α α In the final Table 3, we show min, Pmin, 0, and P′0 with their and that the true value for the slope of 55 Pandora should be error estimates. located towards the right in the error domain. Further Figure 3 shows the enhancement factors ζ versus angular confirmation would require more polarimetric data on these widths d with the error bars (as described above and included asteroids. in Table 2) for the six asteroids. First, for 419 Aurelia with an almost linear phase curve, the distribution is wide as can be CONCLUSION expected. Second, the points for 24 Themis and 1 Ceres are next to one another in agreement with potential similarities in We have succeeded in fitting the representative the composition of the asteroids. Note that 1 Ceres is unusual photometric and polarimetric phase curves for six asteroids of among G-class asteroids: it has a slightly smaller inversion differing taxonomic classes using a joint linear-exponential angle than other G-class asteroids observed (Goidet-Devel model with the angular width of the opposition effect and the et al. 1995). Third, the domain of 55 Pandora is overlapping, negative-polarization branch as the common parameter. It has on one hand, with the domain of 64 Angelina and, on the not been a priori certain that the method would work. We other hand, with the domain of 20 Massalia. There could be conclude that it is probable that common physical mechanisms compositional reasons for a similar opposition effect for these are responsible for the phenomena and that the present study objects: Rivkin et al. (2000) have confirmed hydrated encourages the development of joint physical models for the minerals on the surface of 55 Pandora and placed 55 Pandora phase curves. We stress the importance of acquiring additional in a separate taxonomic class W instead of M. Further studies photometric and polarimetric observations of asteroids to are needed but are beyond the scope of the present study. A extend the present analysis to increasing numbers of objects. systematic trend appears to result, spanning from the low- MCMC methods provide high performance in both albedo to high-albedo asteroids (the albedo being here the separate and joint empirical modeling of the photometric and ). polarimetric phase curves of asteroids. Error estimates follow We plot kI/bI versus d (Fig. 3b), Pmin versus d (Fig. 3c), rapidly and rigorously for both linear and nonlinear and P′0 versus d (Fig. 3d) for the same asteroids. Whereas k / parameters. We envisage application of MCMC to the phase ′ I bI versus d shows no clear trends, Pmin versus d and P0 versus curves of additional asteroids as well as . d show almost linear trends of deepening Pmin and steepening polarimetric slope with increasing d. The F-class asteroid 419 Acknowledgments–The authors express their gratitude to Aurelia appears to provide an intriguing exception to the ISSI for making their collaboration possible as an ISSI linear trend in Pmin versus d. Last but not least, there are hints International Team. K. M.’s research has been partially about a bimodal structure of the patterns, with none of the supported by the EU/TMR project entitled “European present asteroids showing angular widths around 4–4.5°. The Leadership in Space ” (ELSA). A. C. polarimetric parameters of 55 Pandora lie close to those of 20 acknowledges ASI (Contract no. I/015/07/0) for enabling Massalia, with Pmin distinguishing 55 Pandora from 64 his work in Bern. M. D. carried out part of the research Angelina. while he was Henri Poincare’ Fellow at the Observatoire de ′ We further plot Pmin versus kI/bI (Fig. 3e) and P0 versus la Cote d’Azur. The Henri Poincare’ Fellowship is funded kI/bI (Fig. 3f). Within the errors depicted, we can envisage a by the CNRS-INSU, the Conseil General des Alpes- trend for P versus k /b as reported by Shevchenko (1997) Maritimes and the Rotary International—District 1730. A. min ′ I I and a trend for P0 versus kI/bI. For these two trends to be real, C. L. R. acknowledges partial funding from CNES. E. T.’s the true value for the photometric slope of 419 Aurelia should participation in this effort was supported through NASA Asteroid photometric and polarimetric phase curves 1945

Subcontract no. 132416 under NASA JPL prime contract Jeffreys H. 1946. An invariant form for the prior probability in task order no. NMO7108563. estimation problems. Proceedings of the Royal Statistical Society of London, Series A 186:453–461. Kaasalainen S., Piironen J., Kaasalainen M., Harris A. W., Muinonen Editorial Handling—Dr. Nancy Chabot K., and Cellino A. 2003. Asteroid photometric and polarimetric phase curves: Empirical interpretation. Icarus 161:34–46. REFERENCES Levasseur-Regourd A. C. 2003. physical properties. Advances in Space Research 31:2599–2606. Avramchuk V. V., Rosenbush V. K., and Bul’Ba T. P. 2007. Levasseur-Regourd A. C. 2004. Polarimetry of dust in the solar Photometric study of the major satellites of . Solar System system: Remote observations, in situ measurements and Research 41:186–202. experimental simulations. In Photopolarimetry in remote Belskaya I. N. and Shevchenko V. G. 2000. Opposition effect of sensing, edited by Videen G., Yatskiv Y., and Mishchenko M. asteroids. Icarus 147:94–105. Dordrecht: Kluwer Academic Publishers. 393–410. Belskaya I. N., Shevchenko V. G., Efimov Yu. S., Shakhovskoy N. Lumme K. and Muinonen K. 1993. A two-parameter system for M., Shkuratov Yu. G., Gaftonyuk N. M., Gil-Hutton R., Krugly linear polarization of some solar system objects. Asteroids, Yu. N., and Chiorny V. G. 2002. Opposition polarimetry and Comets, Meteors 1993. IAU Symposium vol. 160. p. 194. photometry of the low albedo asteroid 419 Aurelia. In Lumme K., Muinonen K., Harris A. W., and Bowell E. 1993. A new Proceedings of Asteroids, Comets, Meteors 2002. Berlin: ESA three–parameter magnitude system for asteroids. Bulletin of the Publishing Division. pp. 489–491. American Astronomical Society 25:1125. Belskaya I. N., Shevchenko V. G., Kiselev N. N., Krugly Yu., N., Lupishko D. F., Kiselev N. N., Chernova G. P., Shakhovskoj N. M., Shakhovskoy N. M., Efimov Yu. S., Gaftonyuk N. M., Cellino and Vasilyev S. V. 1994. Polarization phase dependences of A., and Gil-Hutton R. 2003. Opposition polarimetry and asteroids 55 Pandora and 704 Interamnia. Kinemat. Phys. Neb. photometry of S and E-type asteroids. Icarus 166:276–284. Tel 10(2):40–44. Belskaya I. N., Shkuratov Yu. G., Emov Yu. S., Shakhovskoy N. M., Mishchenko M. I. and Dlugach J. M. 1993. Coherent backscatter and Gil-Hutton R., Cellino A., Zubko E. S., Ovcharenko A. A., the opposition effect for E-type asteroids. Planetary and Space Bondarenko S. Yu., Shevchenko V. G., Fornasier S., Barbieri C. Science 41:173–181. 2005. The F-type asteroids with small inversion angles of Muinonen K. 1989. Electromagnetic scattering by two interacting polarization. Icarus 178:213–221. dipoles Proceedings of the 1989 URSI Electromagnetic Theory Belskaya I. N., Levasseur-Regourd A. C., Cellino A., Emov Y. S., Symposium. pp. 428–430. Shakhovskoy N. M., Hadamcik E., and Bendjoya P. 2009. Muinonen K. 1990. Light Scattering by Inhomogeneous Media: Polarimetry of main belt asteroids: Wavelength dependence. Backward Enhancement and Reversal of Polarization, Ph.D. Icarus 199:97–105. thesis, University of Helsinki, Finland. Bowell E., Hapke B., Lumme K., Harris A. W., Domingue D., and Muinonen K., Piironen J., Shkuratov Yu. G., Ovcharenko A., and Peltoniemi J. I. 1989. Application of Photometric Models to Clark B. E. 2002a. Asteroid photometric and polarimetric phase Asteroids, in Asteroids II, edited by Binzel R. P., Gehrels T., and effects. In Asteroids III, edited by Bottke W., Binzel R. P., Cellino Matthews M. Tucson: The University of Arizona Press. pp. 524– A., and Paolicchi P. Tucson: The University of Arizona Press. pp. 556. 123–138. Cellino A., Gil Hutton R., Tedesco E. F., Di Martino M., and Brunini Muinonen K., Piironen J., Kaasalainen S., and Cellino A. 2002b. A. 1999. Polarimetric observations of small asteroids: Asteroid photometric and polarimetric phase curves: Empirical preliminary results. Icarus 138:129–140. modeling. Memorie Della Società Astronomica Italiana— Cellino A., Belskaya I. N., Bendjoya P., Di Martino M., Gil-Hutton Journal of the Italian Astronomical Society 73:716–721. R., Muinonen K., and Tedesco E. F. 2006. The strange Muinonen K., Zubko E., Tyynelä J., Shkuratov Yu. G., and Videen G. polarimetric behavior of asteroid (234) Barbara. Icarus 180:565– 2007. Light scattering by Gaussian random particles with 567. discrete-dipole approximation. Journal of Quantitative Chernova G. P., Lupishko D. F., and Shevchenko V. G. 1994. Spectroscopy and Radiative Transfer 106:360–377. Photometry and polarimetry of asteroid 24 Themis. Kinemat. Muinonen K., Cellino A., Belskaya I., Delbo M., Levasseur-Regourd Phys. Neb. Tel. 10(2):45–49. A. C., and Tedesco E. F. 2008. Inversion of asteroid phase curves Gehrels T. 1956. Photometric studies of asteroids: V. The lightcurve for empirical magnitude and polarization systems. In Asteroids, and phase function of 20 Massalia. The Astronomical Journal Comets, Meteors 2008, July 13–18, 2008, Baltimore, Maryland, 123:331–335. USA. LPI Contribution 1405. p. 8263. Gilks W. R., Richardson S., and Spiegelhalter D. J. 1996. Markov Penttilä A., Lumme K., Hadamcik E., and Levasseur-Regourd A. C. Chain Monte Carlo in practice. Chapman and Hall/CRC. 2005. Statistical analysis of asteroidal and cometary polarization Goidet-Devel B., Renard J. B., and Levasseur-Regourd A. C. 1995. phase curves. Astronomy and Astrophysics 432:1081–1090. Polarization of asteroids: Synthetic curves and characteristic Rivkin A., Howell E. S., Lebofsky L. A., Clark B. E., and Britt D. T. parameters. Planetary and Space Science 43:779–786. 2000. The nature of M-class asteroids from 3-µm observations. Hapke B. 1990. Coherent backscatter and the radar characteristics of Icarus 145:351–368. outer satellites. Icarus 88:407–417. Rosenbush V., Kiselev N., Avramchuk V., and Mishchenko M. 2002. Harris A. W., Young J. W., Bowell E., Martin L. J., Millis R. L., Photometric and polarimetric opposition phenomena exhibited Poutanen M., Scaltriti F., Zappalà, V., Schober H. J., Debehogne by solar system bodies. In Optics of cosmic dust, edited by H., and Zeigler K. W. 1989a. Photoelectric observation of Videen G. and Kocifaj M. NATO Science Series. Dordrecht/ asteroids 3, 24, 60, 261, 863. Icarus 77:171–186. Boston/London: Kluwer Academic Publishers. pp. 191–226. Harris A. W., Young J. W., Contreiras L., Dockweiler T., Belkora L., Rosenbush V. K., Kiselev N. N., Shevchenko V. G., Jockers K., Salo H., Harris W. D., Bowell E., Poutanen M., Binzel R. P., Shakhovskoy N. M., and Efimov Yu. S. 2005. Polarization and Tholen D. J., and Wang S. 1989b. Phase relations of high albedo brightness opposition effects for the E-type asteroid 64 Angelina. asteroids: The unusual opposition brightening of and 64 Icarus 178:222–234. Angelina. Icarus 81:365–374. Rosenbush V. K., Shevchenko V. G., Kiselev N. N., Sergeev 1946 K. Muinonen et al.

A. V., Shakhovskoy N. M., Velichko F. P., Kolesnikov S. V., polarization of light scattered by the surfaces of atmosphereless and Karpov N. V. 2009. Polarization and brightness celestial bodies. Astronomicheskii Vestnik 23:176–180. opposition effects for the E-type Asteroid 44 Nysa. Icarus Tedesco E. F., Taylor R. C., Drummond J., Harwood D., Nickoloff I., 201:655–665. Shevchenko V. G., Krugly Yu. N., Lupishko D. Scaltriti F., Schober H. J., and Zappalà, V. 1983. Worldwide F., Harris A. W., and Chernova G. P. 1993. Lightcurves and photometry and lightcurve observations of 1 Ceres during the phase relations of asteroid 55 Pandora. Astronomicheskii 1975–1976 apparition. Icarus 54:23–29. Vestnik 27(3):75–80. Tyynelä J., Zubko E., Videen G., and Muinonen K. 2007. Interrelating Shevchenko V. G. 1997. Analysis of asteroid brightness-phase angular scattering characteristics to internal electric fields for relations. Solar System Research 31:219–224. wavelength-scale spherical particles. Journal of Quantitative Shkuratov Yu. G. 1985. On the origin of the opposition effect and Spectroscopy and Radiative Transfer 106:520–534. negative polarization for cosmic bodies with solid surface. Zellner B. and Gradie J. 1976. Minor and related objects. XX. Astronomical Circular No. 1400, Sternberg State Astronomy Polarimetric evidence for the and compositions of 94 Institute, Moscow. pp. 3–6. asteroids. The Astronomical Journal 81:262–280. Shkuratov Yu. G. 1988. Diffractional model of the brightness surge of Zellner B., Gehrels T., and Gradie J. 1974. Minor planets and related complex structures. Kin. Phys. Neb. Tel. 4:60–66. objects. XVI. Polarimetric diameters. The Astronomical Journal Shkuratov Yu. G. 1989. A new mechanism of the negative 79:1100–1110.